A framed $f$-structure on the tangent bundle of a Finsler manifold

Esmaeil Peyghan¹ ; Chunping Zhong²

¹ Department of Mathematics Faculty of Science Arak University Arak 38156-8-8349, Iran
² School of Mathematical Sciences Xiamen University Xiamen 361005, China

Annales Polonici Mathematici, Tome 104 (2012) no. 1, pp. 23-41

Résumé

Let $(M,F)$ be a Finsler manifold, that is, $M$ is a smooth manifold endowed with a Finsler metric $F$. In this paper, we introduce on the slit tangent bundle $\widetilde{TM}$ a Riemannian metric $\widetilde{G}$ which is naturally induced by $F$, and a family of framed $f$-structures which are parameterized by a real parameter $c\neq 0$. We prove that (i) the parameterized framed $f$-structure reduces to an almost contact structure on $IM$; (ii) the almost contact structure on $IM$ is a Sasakian structure iff $(M,F)$ is of constant flag curvature $K=c;$ (iii) if $\mathcal{S}=y^i\delta_i$ is the geodesic spray of $F$ and $R(\cdot,\cdot)$ the curvature operator of the Sasaki–Finsler metric which is induced by $F$, then $R(\cdot,\cdot)\mathcal{S}=0$ iff $(M,F)$ is a locally flat Riemannian manifold.

Détail

DOI : 10.4064/ap104-1-3

Keywords: finsler manifold smooth manifold endowed finsler metric paper introduce slit tangent bundle widetilde riemannian metric widetilde which naturally induced family framed f structures which parameterized real parameter neq prove nbsp parameterized framed f structure reduces almost contact structure nbsp nbsp almost contact structure sasakian structure constant flag curvature iii nbsp mathcal delta geodesic spray cdot cdot curvature operator sasaki finsler metric which induced cdot cdot mathcal locally flat riemannian manifold